Title: The Hodge theory of algebraic maps

Abstract: I will report on my joint work with Luca Migliorini at Bologna on the homology of algebraic maps. We give a geometric proof of the so-called Decomposition Theorem due to Beilinson-Bernstein-Deligne and Gabber concerning the relation between the homology theories on the domain and target of an algebraic map. Our approach identifies the non degeneration of certain intersection forms on the homology of the fibers of the map as the reason for the decomposition of the homology of the domain into certain pieces defined on the target. This fact has no counterpart in other geometries, e.g. complex geometry, real algebraic geometry. I will illustrate the discussion with some key elementary examples.